统计与数据科学学院知识库

空间结构数据分析与统计建模问题一直是许多科学领域研究的热点话题。简单的全局回归建模不仅忽略空间地理位置的数据信息，而且当协变量存在空间非平稳效应时无法满足研究的需要。空间变系数模型(SVCMs)是一种研究空间非平稳数据的有效工具，它可以用来研究空间依赖性和空间非平稳性质的影响关系。近年来，许多文献致力于探讨均值意义下的SVCMs。然而，由于数据产生机制的复杂性，均值回归方法高斯误差的强制假设在实际生活中很难满足。分位回归估计方法会克服这一缺陷。所以将均值回归推广至分位回归很有必要。

首先，本文基于空间变系数非参数回归模型，研究了的一种基于二元惩罚样条逼近的分位回归估计方法，该估计方法不仅可以处理具有复杂边界、不规则形状的空间区域，而且还展现出不同分位水平下的解释能力。在两种不同的情形下，分别给出了所提出估计量的理论性质，即收敛速率和渐近分布。对于参数的估计过程，提出一种基于交替方向乘子(ADMM)迭代算法实现模型的求解。对于模型拟合优度检验问题，提出了基于Boostrap方法的拟合优度检验方法，并给出了检验实现算法。基于复合分位回归的良好性能，也提出了该模型的二元惩罚样条逼近的复合分位回归估计量和其实现算法。数值模拟结果显示所提出的估计方法比均值意义下更稳健。

其次，考虑到空间数据的平稳效应和非平稳效应共存，提出空间部分线性变系数模型(SPLVCMs)，它是一种半参数模型，其线性部分和非参数部分分别表示空间协变量的平稳性和非平稳性影响。对于模型参数估计，提出了基于二元惩罚样条的非参数逼近分位回归估计方法，研究了其线性部分与非参数部分估计量的理论性质。针对模型实现，提出了基于ADMM的参数估计算法。不同情形下的仿真模拟显示所提出的模型和方法在非正态误差下有明显的优势，估计结果更加稳健、有效。

最后，利用佛罗里达州的失业率实际数据和我国空气质量数据验证本文所提出方法的应用价值。结果表明，所提出的模型能充分刻画协变量和响应变量之间的非平稳性影响关系，可为后续应用研究提供新的方法理论。

Analysis and statistical modeling of spatial structure data has always been a hot topic in many scientific fields. The simple global regression modeling not only ignores the data information of spatial geographic location, but also fails to meet the needs of research when the covariables have spatial non-stationary effects. Spatial Varying Coefficient Models(SVCMs) is an effective tool to study spatial nonstationary data.  It can be used to study the relationship between spatial dependence and spatial nonstationary properties. In recent years, the research on SVCMs in the meaning of mean value is common. However, due to the complexity of data generation mechanism, the mandatory assumption of Gaussian error in mean regression method is difficult to satisfy in real life. The quantile regression estimation method can overcome this defect. Therefore, it is necessary to generalize mean regression to quantile regression estimation.

At First, this thesis introduces the spatial varying coefficient nonparametric regression model, and develops a quantile regression estimation method based on bivariate penalty spline approximation. This estimation method can not only deal with the spatial region with complex boundary and irregular shape, but also shows the interpretation ability under different quantile levels. In two different cases, the theoretical properties of the proposed estimator, the rate of convergence and asymptotic distribution, are given respectively. For the estimation process of parameters, an iterative algorithm based on Alternating Direction Multiplier Method (ADMM) is proposed to solve the model. To test the goodness-of-fit of the model, a goodness-of-fit test method based on Boostrap method is proposed in this thesis, and the test implementation algorithm is given. Based on the good performance of composite quantile regression, the thesis also presents the composite quantile regression estimator of bivariate penalty spline approximation of the model and its implementation algorithm.The numerical simulation results show that the proposed estimation method is more robust than the mean value.

Secondly, considering the coexistence of stationary effect and non-stationary effect of spatial data, the Spatial Partial Linear Varying Coefficient Models(SPLVCMs) is proposed. This model is a semi-parametric model, including linear part and non-parametric part, which represent the stationarity and non-stationarity effect of spatial covariables respectively. For the parameter estimation, a non-parametric approximation quantile regression estimation method based on bivariate penalty splines is proposed, and the theoretical properties of the linear part and the non-parametric part estimator are studied. A parameter estimation algorithm based on ADMM is proposed to realize the model. In the numerical simulation stage, the simulation results under different conditions show that the proposed method has obvious advantages under non-normal error, and the estimation results are more robust and effective.

Finally, this thesis uses actual data of Florida unemployment rate and air quality data of China to verify the application value of the proposed method. The results show that the proposed models can fully describe the nonstationary influence relationship between covariates and response variables, which can provide a new method and theory for subsequent application research.